The four-point obstruction conjecture for labeled star graphs

Let $(X^*,d^*)$ be a finite ultrametric space. Write ${\bf US}$ for the class of finite ultrametric spaces generated by labeled star graphs, and say that two ultrametric spaces are weakly similar if there is a bijection between them whose distances are related by a strictly increasing function. The four-point spaces $(X_4,d_4)$ and $(Y_4,\rho_4)$ are the spaces described in Figure 3 of the source.

Four-point obstruction conjecture. The following statements are equivalent:

$$(X^*,d^*)\notin {\bf US}.$$ $$(X^*,d^*)\text{ contains a four-point subspace weakly similar either to }(X_4,d_4)\text{ or to }(Y_4,\rho_4).$$

If true, this would give a finite forbidden-subspace characterization of ultrametric spaces generated by labeled star graphs. The source presents the equivalence as a conjecture and gives the two four-point spaces as basic obstructions; no resolution is supplied here.